%\setcounter{chapter}{7}
\chapter{Quantum noise and quantum operations}
\Textbf{8.1}
Density operator of initial state is written by $\kb{\psi}$ and final state is written by $U\kb{\psi}U^\dagger$.
Thus time development of $\rho = \kb{\psi}$ can be written by $\mathcal{E}(\rho) = U\rho U^\dagger$.

\Textbf{8.2}
From eqn (2.147) (on page 100),
\begin{align*}
	\rho_m = \frac{M_m \rho M_m^\dagger}{\Tr (M_m^\dagger  M_m\rho ) }
					= \frac{M_m \rho M_m^\dagger}{\Tr ( M_m \rho M_m^\dagger ) }
					= \frac{\mathcal{E}_m (\rho)}{\Tr \mathcal{E}_m (\rho)}.
\end{align*}

And from eqn (2.143) (on page 99), $p(m) = \Tr (M_m^\dagger M_m \rho) = \Tr (M_m \rho M_m^\dagger) = \Tr \mathcal{E}_m (\rho)$.


\Textbf{8.3}



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